In other Eulerian news (how often do you get to use that phrase?), I just finished reading Euler: The Master of Us All by Wiliam Dunham, who is interested in the history of mathematics. I'd read his previous book Journey through Genius: The Great Theorems of Mathematics when I was in high school; this book treats a dozen or so of the major theorems of mathematics, one to a chapter, in a way that's probably accessible to the bright high school student. His book on Euler is a bit trickier -- it assumes one is familiar with the calculus and comfortable with infinite series. Euler was very comfortable with infinite series; this is what enabled, for example, his solution of the Basel problem: finding the sum 1 + 1/4 + 1/9 + 1/16 + ..., which turns out to be π2/6. The Wikipedia article gives two proofs, Euler's proof and a more rigorous one. Personally, I prefer Euler's proof; I often find that the rigor of modern proofs obscures why the result in question is actually true.
Dunham's book treats some of Euler's major results:
- the Euclid-Euler theorem on perfect numbers: all even perfect numbers have the form (2k-1)2k-1, where 2k-1 is prime. Odd perfect numbers are harder to find; Carl Pomerance has a heuristic argument that they don't exist.
- infinite series involving logarithms; apparently Euler was the first to look at logarithms as the inverse of the exponential function, as opposed to just a useful computational aid.
- the solution to the Basel problem and some related series; in fact Dunham talked about this in Journey through Genius as well, which is where I first learned about it (edit, 7/13/07: you can read more about the Basel problem, which is the summing of the series 1 + 1/4 + 1/9 + 1/16 + ... = π2/6, at the Everything Seminar
- the solution of third- and fourth-degree algebraic equations. (Did you know that any quartic polynomial with real coefficients can be factored as the product of two quadratics with real coefficients? I didn't.)
- some old-fashioned synthetic geometry: Heron's formula for the area of a triangle, for example
- solutions of the derangement problem and the fact that partitions into odd parts and into distinct parts are equinumerous
1 + 1/2 + 1/3 + 1/4 + 1/5 + ... = (2 x 3 x 5 x 7 x 11 x 13 x ...)/(1 x 2 x 4 x 6 x 10 x 12 x ...)
which seems silly, because both sides are infinite. But it makes sense. We have, for example,
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2/1
1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ...
= (1 + 1/2 + 1/4 + 1/8 + ...) (1 + 1/3 + 1/9 + 1/27 + ...)
= (2/1) (3/2)
where the original sum is the sum of reciprocals of numbers whose only prime factors are 2 and 3. Why not extend this to the sum of the reciprocals of all integers using unique factorization? And it turns out that this sum has a perfectly reasonable interpretation: by the same sort of argument as above,
1 + 1/2s + 1/3s + 1/4s + ... = 1/[(1-2-s)(1-3-s)(1-5-s)] ...
where if s>1 both sides converge; if you let s approach 1 from above you recover Euler's result. (This is due to Kronecker, and can be found on pp. 66-70 of Dunham.) But this sort of heuristic appeals to me. It seems to me that modern mathematicians don't take enough risks like this. Or, if they do, they are not telling me; I fear that the papers currently being published omit anything which isn't totally rigorous, even if (especially if?) it might make things clearer. I suppose this makes sense if you want to keep your papers short, which was of course important when papers were printed on, well, paper -- but these sorts of less rigorous things could certainly be made available in some sort of supplement to one's paper, published on a web page, for example.
Finally, given how many areas Euler has touched, sometimes I've joked that we should just rename mathematics "Euler". Somewhat more seriously, though -- I wonder to what extent one can tell whether someone is a mathematician by writing the word "Euler" on a piece of paper and asking them to pronounce it. (Isaac Asimov, who was trained as a chemist, suggested that "unionized" played the same role for chemists. Chemists pronounce it in four syllables and think it refers to molecules which have the same number of protons as electrons and thus a neutral charge; non-chemists pronounce it in three syllables and think it refers to organized labor.)
(My apologies for the awkwardness of the notation here. There seem to be ways to write out mathematics in blog posts -- either using LaTeX or MathML -- but they're not necessary for the notation I want to use here and seem like they'd take a little fooling around to get to work. And I don't even know MathML, although I wouldn't mind learning it.)